A Functorial Set assigns to each object of a category a set, while morphisms induce structure-preserving maps between these sets. A Functorial Structure is a covariant functor into Set, where the functor-assigned elements are treated as structured objects equipped with natural pushforwards 1, 2. In this paper, we investigate the notions of Functorial Number, Functorial Graph, and Functorial HyperGraph. A Functorial Number is defined as a functor that assigns to each object of a category a semiring, with morphisms inducing structurepreserving homomorphisms. A Functorial Graph is a covariant functor that assigns graphs to objects and graph homomorphisms to morphisms, preserving identities and compositions. A Functorial HyperGraph is a covariant functor that assigns hypergraphs to objects and hypergraph homomorphisms to morphisms, functorially preserving the structural relationships throughout.
Takaaki Fujita (Tue,) studied this question.